\section{Invalidation of previous results and misconceptions}
\label{sec:example}

A \textit{critical instant} for a task $\tau_i$ is defined as an instant at which a request for that task will have the largest response time. Since the response time of a task is dependent on the higher priority tasks, a critical instant for a task $\tau_i$ is generally concerned with the release pattern of higher priority tasks. 
In~\cite{Karthik:RTAS10}, Lakshmanan et. al. argue that the release pattern $\ssPhi$ is a critical instant for a self-suspending task $\sstask$, where $\ssPhi$ is defined as follows:
\begin{itemize}
	\item every higher-priority non-self-suspending task $\tau_h \equals \left\langle \left(C_{h}\right), D_h, T_h\right\rangle$ is released simultaneously with $\sstask$;
	\item jobs of $\tau_h$ eligible to be released during any $j$th ($1 \leq j \leq m_i$) suspending region of $\sstask$ are delayed to be aligned with the release of subsequent $(j+1)$th computing region of $\sstask$; and
	\item all remaining jobs of $\tau_h$ are released every $T_h$.
\end{itemize}

We now show with an example that $\ssPhi$ is not a critical instant for a self-suspending task $\sstask$.
\begin{Example}
Consider a task set $\tau = \left\{\tau_1, \tau_2, \tau_3\right\}$ of three constrained-deadline self-suspending sporadic tasks scheduled on a single processor. Let the characteristics of these tasks be as follows: $\tau_1 \equals \left\langle \left(1\right), 4, 4\right\rangle$; $\tau_2 \equals \left\langle \left(1\right), 100, 100\right\rangle$ and $\tau_3 \equals \left\langle \left(1, 2, 3\right), 1000, 1000\right\rangle$. Let the priorities of tasks be assigned using the Rate Monotonic policy (i.e., smaller the period, higher the priority); this implies that task $\tau_1$ has the highest priority, task $\tau_2$ has the medium priority and task $\tau_3$ has the lowest priority. Let us compute the response time of task $\tau_3$ considering two different job release patterns: (i) a job release pattern $\Phi_3$ and (ii) a job release pattern different than $\Phi_3$. We show that there exists a job release pattern which \textit{is not} $\Phi_3$ and for such a job release pattern, the response-time of task $\tau_3$ is higher than its response time when the job release pattern is $\Phi_3$.

\textit{Scenario 1: for the job release pattern $\Phi_3$.} Let us consider the job release pattern $\Phi_3$ as shown in Fig.~\ref{fig:ex-phi}.
\begin{figure}
  \centering
  \subfloat[Scenario 1. Response-time analysis when the job release pattern \textit{is} $\Phi_3$.]{\label{fig:ex-phi} \includegraphics[height=2.6cm, width=0.45\textwidth]{ex-phi}} \\
  \subfloat[Scenario 2. Response-time analysis when the job release pattern \textit{is not} $\Phi_3$.]{\label{fig:ex-no-phi} \includegraphics[height=2.6cm, width=0.45\textwidth]{ex-no-phi}}
  \caption{An example to show that $\ssPhi$ is not the critical instant of task $\sstask$.}
  \label{fig:hist-comp}
\end{figure}
Using the standard response-time expression, we obtain the response time $R_{3,1}=3$ for the computing region $\tau_{3,1}$ and $R_{3,2}=4$ for the computing region $\tau_{3,2}$.
%Upon determining the response time $R_{3,1}$ of the computing region $\tau_{3,1}$ using the standard response-time expression, we obtain: $R_{3,1} = 3$. 
%With this, the second job of task $\tau_1$ in Fig.~\ref{fig:ex-phia} is released during the suspending region of task $\tau_3$. In order to obtain the job release pattern $\Phi_3$, let us (i) delay the release of the second job of task $\tau_1$ by one time unit (i.e., it is released at time $5$ now instead of $4$) so that it is now released at the same time as the computing region $\tau_{3,2}$ and (ii) delay all the subsequent job releases of $\tau_1$ by one time unit as well, in order to respect its minimum inter-arrival time of $\tau_1$. The job release pattern $\Phi_3$ is shown in Fig.~\ref{fig:ex-phib}. 
%And upon determining the response time $R_{3,2}$ of the computing region $\tau_{3,2}$, we obtain: $R_{3,2} = 4$. 
Hence, for this scenario, we obtain the response-time of task $\tau_3$ to be: $R_3~=~R_{3,1}+S_1+R_{3,2}~=~3+2+4~=~9$.

\textit{Scenario 2: for a job release pattern different than $\Phi_3$.} Let us consider a job release pattern as shown in Fig.~\ref{fig:ex-no-phi}. Observe that this release pattern is not $\Phi_3$ since the task $\tau_2$ is not simultaneously released with task $\tau_1$. For this scenario, we obtain: $R_{3,1}=2$ for the computing region $\tau_{3,1}$ and $R_{3,2}=6$ for the computing region $\tau_{3,2}$. Hence, for this scenario, the response-time of task $\tau_3$ is given by: $R_3~=~2+2+6~=~10$.

Clearly, the response-time of task $\tau_3$ obtained in Scenario~2 is higher than the response-time of $\tau_3$ obtained in Scenario~1. Hence, the claim of Lakshmanan et. al.~\cite{Karthik:RTAS10} that $\ssPhi$ is the critical instant for a self-suspending task $\sstask$ is incorrect.
\qed
\end{Example}

